## Tohoku Mathematical Journal

### Small noise asymptotic expansions for stochastic PDE's, I. The case of a dissipative polynomially bounded non linearity

#### Abstract

We study a reaction-diffusion evolution equation perturbed by a Gaussian noise. Here the leading operator is the infinitesimal generator of a $C_0$-semigroup of strictly negative type, the nonlinear term has at most polynomial growth and is such that the whole system is dissipative.

The corresponding Itô stochastic equation describes a process on a Hilbert space with dissipative nonlinear, non globally Lipschitz drift and a Gaussian noise.

Under smoothness assumptions on the nonlinearity, asymptotics to all orders in a small parameter in front of the noise are given, with uniform estimates on the remainders. Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular example we consider the small noise asymptotic expansions for the stochastic FitzHugh-Nagumo equations of neurobiology around deterministic solutions.

#### Article information

Source
Tohoku Math. J. (2), Volume 63, Number 4 (2011), 877-898.

Dates
First available in Project Euclid: 6 January 2012

https://projecteuclid.org/euclid.tmj/1325886292

Digital Object Identifier
doi:10.2748/tmj/1325886292

Mathematical Reviews number (MathSciNet)
MR2872967

Zentralblatt MATH identifier
1234.35328

#### Citation

Albeverio, Sergio; Di Persio, Luca; Mastrogiacomo, Elisa. Small noise asymptotic expansions for stochastic PDE's, I. The case of a dissipative polynomially bounded non linearity. Tohoku Math. J. (2) 63 (2011), no. 4, 877--898. doi:10.2748/tmj/1325886292. https://projecteuclid.org/euclid.tmj/1325886292

#### References

• S. Albeverio and C. Cebulla, Synchronizability of stochastic network ensembles in a model of interacting dynamical units, Physica A Stat. Mech. Appl 386 (2007), 503–512.
• S. Albeverio, L. Di Persio and E. Mastrogiacomo, Invariant measures for stochastic differential equations on networks, in preparation.
• S. Albeverio, V. Fatalov and V. I. Piterbarg, Asymptotic behavior of the sample mean of a function of the Wiener process and the Macdonald function, J. Math. Sci. Univ. Tokyo 16 (2009), 55–93.
• S. Albeverio, F. Flandoli and Y. G. Sinai, SPDE in hydrodynamic: recent progress and prospects, Lectures given at the C.I.M.E. Summer School held in Cetraro, August 29–September 3, 2005, Edited by G. Da Prato and M. Röckner, Lecture Notes in Mathematics, vol. 1942, Springer-Verlag, Berlin, 2008.
• S. Albeverio and S. Liang, Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables, Ann. Probab. 33 (2005), 300–336.
• S. Albeverio, Wiener and Feynman-path integrals and their applications, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994), pp. 153–194. Proc. Sympos. Appl. Math. 52, Amer. Math. Soc., Providence, RI, 1997.
• S. Albeverio, E. Lytvynov and A. Mahnig, A model of the term structure of interest rates based on Lévy fields, Stochastic Process. Appl. 114 (2004), 251–263
• S. Albeverio and S. Mazzucchi, The trace formula for the heat semigroup with polynomial potential, Proc. Seminar Stochastic Analysis, Random Fields and Applications VI, Ascona 2008, (Edited by R. Delang, M. Dozzi, F. Russo.), pp. 3–22, Birkhäuser, Basel, 2011.
• S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields 89 (1991), 347–386.
• S. Albeverio, H. Röckle and V. Steblovskaya, Asymptotic expansions for Ornstein-Uhlenbeck semigroups perturbed by potentials over Banach spaces, Stochastics Stochastics Rep. 69 (2000), 195–238.
• S. Albeverio and V. Steblovskaya, Asymptotics of infinite-dimensional integrals with respect to smooth measures. I, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), 529–556.
• S. Bonaccorsi and E. Mastrogiacomo, Analysis of the stochastic FitzHugh-Nagumo system, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), 427–446.
• S. Cerrai, Differentiability of Markov semigroups for stochastic reaction-diffusion equations and applications to control, Stochastic Process. Appl. 83 (1999), 15–37.
• S. Cerrai and M. Freidlin, Smoluchowski-Kramers approximation for a general class of SPDEs, J. Evol. Equ. 6 (2006), 657–689.
• S. Cardanobile and D. Mugnolo, Analysis of a FitzHugh-Nagumo-Rall model of a neuronal network, Math. Methods Appl. Sci. 30 (2007), 2281–2308.
• G. Da Prato and L. Tubaro, Self-adjointness of some infinite-dimensional elliptic operators and application to stochastic quantization, Probab. Theory Related Fields 118 (2000), 131–145.
• G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, 1992.
• G. Da Prato and J. Zabczyk, Ergodicity for infinite-dimensional systems, London Mathematical Society Lecture Note Series 229, Cambridge University Press, Cambridge, 1996.
• J.-D. Deuschel and D. W. Stroock, Large deviations, Pure and Applied Mathematics 137, Academic Press Inc., Boston, MA, 1989.
• A. Dembo and O. Zeitouni, Large deviations techniques and applications, Applications of Mathematics 38, Second edition, Springer-Verlag, New York, 1998.
• B. Forster, E. Lütkebohmert and J. Teichmann, Absolutely continuous laws of jump-diffusions in finite and infinite dimensions with applications to mathematical finance, SIAM J. Math. Anal. 40 (2008/09), 2132–2153.
• Y. Inahama and H. Kawabi, Asymptotic expansions for the Laplace approximations for Itô functionals of Brownian rough paths, J. Funct. Anal. 243 (2007), 270–322.
• Y. Inahama and H. Kawabi, On the Laplace-type asymptotics and the stochastic Taylor expansion for Itô functionals of Brownian rough paths, Proceedings of RIMS Workshop on Stochastic Analysis and Applications, 139–152, RIMS Kôkyûroku Bessatsu, B6, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.
• N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library 24, Second edition, North-Holland Publishing Co., Amsterdam, 1989.
• G. Jona-Lasinio and P. K. Mitter, Large deviation estimates in the stochastic quantization of $\phi^4_2$, Comm. Math. Phys. 130 (1990), 111–121.
• G. Jona-Lasinio and P. K. Mitter, On the stochastic quantization of field theory, Comm. Math. Phys. 101 (1985), 409–436.
• A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis. Vol. 2: Measure. The Lebesgue integral. Hilbert space, Translated from the first (1960) Russian ed. by H. Kamel and H. Komm, Graylock Press, Albany, N.Y., 1961.
• K. Lehnertz, J. Arnhold, P. Grassberger and C. E. Elger, Chaos in Brain ?, World Scientific, Singapore, 2000.
• G. E. Ladas and V. Lakshmikantham, Differential equations in abstract spaces, Mathematics in Science and Engineering 85, Academic Press, New York, 1972.
• R. Marcus, Parabolic Itô equations, Trans. Amer. Math. Soc. 198 (1974), 177–190.
• R. Marcus, Parabolic Itô equations with monotone nonlinearities, J. Funct. Anal. 29 (1978), 275–286.
• P. Malliavin and S. Taniguchi, Analytic functions, Cauchy formula, and stationary phase on a real abstract Wiener space, J. Funct. Anal. 143 (1997), 470–528.
• G. Parisi, Statistical field theory, Frontiers in Physics 66, With a foreword by David Pines, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, MA, 1988.
• C. Rovira and S. Tindel, Sharp Laplace asymptotics for a parabolic SPDE, Stochastics Stochastics Rep. 69 (2000), 11–30.
• M. Schilder, Some asymptotic formulas for Wiener integrals, Trans. Amer. Math. Soc. 125 (1966), 63–85.
• B. Simon, Functional integration and quantum physics, Second edition, AMS Chelsea Publishing, Providence, RI, 2005.
• H. C. Tuckwell, Analytical and simulation results for the stochastic spatial FitzHugh-Nagumo model neuron, Neural Comput. 20 (2008), 3003–3033.
• H. C. Tuckwell, Introduction to theoretical neurobiology. Vol. 1, Linear cable theory and dendritic structure, Cambridge Studies in Mathematical Biology 8, Cambridge University Press, Cambridge, 1988.
• H. C. Tuckwell, Introduction to theoretical neurobiology. Vol. 2, Nonlinear and stochastic theories, Cambridge Studies in Mathematical Biology 8, Cambridge University Press, Cambridge, 1988.
• H. C. Tuckwell, Random perturbations of the reduced FitzHugh-Nagumo equation, Phys. Scripta 46 (1992), 481–484.
• S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab. 15 (1987), 1–39.